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WORK
16'
(a)   Horizontal tracks and no resistance.
(b)   Horizontal tracks and a resistance of 12 pounds per ton.
(c)  Down a grade of 1 in 200 with, no resistance.
(d)  Same as in (c) but with a resistance of 12 pounds per ton.
(e)  Up a grade of 1 in 200 with no resistance.
(f)   Same as in (b) but with a resistance of 12 pounds per ton.
10.   A mass of 5 pds. is at the bottom of a vertical shaft which reache the center of the earth.   How much work will have to be done in orde to bring it to the surface?   The weight of a body varies, within the earth directly as its distance from the center.   Take 4000 miles to be the deptl of the shaft.
11.   Express the result of the last problem in joules.
132. Work Done by a Torque.  Suppose the rigid body A Fig. 94, to be given an angular displacement de about a fixec axis through the point 0, perpendicular to the plane of the paper. The displacement may be considered to be] due to a single force which forms a couple with the reaction of the axis, or it may be considered to be due to small forces acting upon every element of mass of the body.
Taking the latter view, let dF be the resultant force* acting upon the element of mas* dm. Then since dm can move only at right angles to th< line r, which joins it to the axis, dF must be perpendiculai to r. When the body is given an angular displacement de dm is displaced through ds, therefore the work done by dF is
d*W = dF  ds = dF  r dd = dG de,
* d F is the resultant of the external forces which act directly on dm, anc of the internal forces which are due to the connection of dm with the rest oi the body.
FIG. 94.